Sampling Distributions

Note: In this chapter we learn sampling distributions.

1. First we will look at a simple activity like example.

2. We will have sections named “DETOUR #”, we will learn some brand name distributions in these sections.

Let’s begin…

1 Sampling Distribution of the sample proportion

1.1 What proportion of this bowl’s balls are red?

Take a look at the bowl in the following Figure. It has a certain number of red and a certain number of white balls all of equal size. Furthermore, it appears the bowl has been mixed beforehand as there does not seem to be any particular pattern to the spatial distribution of red and white balls.

Let’s now ask ourselves, what proportion of this bowl’s balls are red?

One way to answer this question would be to perform an exhaustive count: remove each ball individually, count the number of red balls and the number of white balls, and divide the number of red balls by the total number of balls. However this would be a long and tedious process.

Observe that ____ of the balls are red and there are a total of ____ balls and thus ___ % of the shovel’s balls are red. We can view the proportion of balls that are red in this shovel as a guess of the proportion of balls that are red in the entire bowl. While not as exact as doing an exhaustive count, our guess of ___% took much less time and energy to obtain.

However, say, we started this activity over from the beginning. In other words, we replace the 50 balls back into the bowl and start over. Would we remove exactly 17 red balls again? In other words, would our guess at the proportion of the bowl’s balls that are red be exactly 34% again? Maybe?

What if we repeated this exercise several times? Would I obtain exactly 17 red balls each time? In other words, would our guess at the proportion of the bowl’s balls that are red be exactly 34% every time? Surely not.

Let’s try do this on the computer…

To this end, we use a data frame bowl in the moderndive package whose rows correspond exactly with the contents of the actual bowl.

head(bowl)

# A tibble: 6 x 2
  ball_ID color
    <int> <chr>
1       1 white
2       2 white
3       3 white
4       4 red  
5       5 white
6       6 white

# View(bowl) # Use this in the console

Observe in the output that bowl has ___ rows, telling us that the bowl contains ___ equally-sized balls. The first variable ball_ID is used merely as an “identification variable”, none of the balls in the actual bowl are marked with numbers. The second variable color indicates whether a particular virtual ball is red or white.

Now that we have a virtual analogue of our bowl, we now need a virtual analogue for the shovel seen in Figure 2; we’ll use this virtual shovel to generate our virtual random samples of 50 balls. We’re going to use the rep_sample_n() function included in the moderndive package. This function allows us to take repeated, or replicated, samples of size n. Run the following and explore.

virtual_shovel <- bowl %>% 
  rep_sample_n(size = 50)

virtual_shovel

# A tibble: 50 x 3
# Groups:   replicate [1]
   replicate ball_ID color
       <int>   <int> <chr>
 1         1     136 red  
 2         1     673 white
 3         1    1819 red  
 4         1     627 red  
 5         1     438 white
 6         1     510 white
 7         1     628 white
 8         1    1473 red  
 9         1     870 white
10         1    1636 white
# … with 40 more rows

Next we can find out how many res ones are there in our virtual_shovel

virtual_shovel %>% 
  summarize(num_red = sum(color=="red"))  

# A tibble: 1 x 2
  replicate num_red
      <int>   <int>
1         1      23

How about the proportion on red? We can use the mutate (new) function to create a new variable, in this case prop_red.

virtual_shovel %>% 
  summarize(num_red = sum(color == "red")) %>% 
  mutate(prop_red = num_red / 50)

# A tibble: 1 x 3
  replicate num_red prop_red
      <int>   <int>    <dbl>
1         1      23     0.46

1.2 Using the virtual shovel many times

virtual_samples <- bowl %>% 
  rep_sample_n(size = 50, reps = 30)

kable(virtual_samples)

replicate	ball_ID	color
1	2313	red
1	1783	red
1	1096	white
1	2095	white
1	2324	white
1	1765	red
1	741	white
1	668	red
1	980	white
1	1292	red
1	2030	white
1	1225	white
1	1078	white
1	1873	red
1	1525	white
1	1799	white
1	1706	white
1	1128	white
1	5	white
1	111	white
1	331	red
1	2392	red
1	1804	red
1	946	white
1	2042	red
1	2058	red
1	723	white
1	1808	red
1	1531	red
1	1759	white
1	1809	white
1	1219	white
1	501	red
1	1552	white
1	2369	red
1	876	red
1	27	red
1	1592	white
1	78	white
1	1630	white
1	1342	white
1	2342	red
1	1774	red
1	1886	white
1	1333	white
1	771	white
1	441	white
1	1876	white
1	2215	white
1	250	red
2	1328	red
2	610	white
2	1333	white
2	40	red
2	2112	white
2	1907	white
2	1524	white
2	2347	white
2	1470	white
2	1450	red
2	212	red
2	589	red
2	2175	white
2	2070	white
2	725	white
2	1850	white
2	1199	white
2	2389	white
2	553	white
2	1261	white
2	2351	red
2	1388	white
2	222	red
2	2099	white
2	395	red
2	1736	white
2	2229	white
2	1910	white
2	1463	red
2	2282	red
2	407	red
2	986	red
2	1722	white
2	2088	white
2	741	white
2	2305	red
2	1951	white
2	474	red
2	1098	red
2	1476	red
2	542	white
2	2189	white
2	2103	white
2	289	red
2	1567	red
2	1126	white
2	1851	red
2	2192	red
2	979	white
2	953	red
3	1088	red
3	330	white
3	2369	red
3	1724	white
3	1190	red
3	1861	white
3	227	red
3	845	white
3	1425	red
3	1528	white
3	57	white
3	34	white
3	1998	red
3	1906	red
3	1504	red
3	1172	red
3	581	red
3	2061	white
3	2281	white
3	339	red
3	842	red
3	2057	red
3	1829	white
3	738	white
3	800	red
3	1268	white
3	655	white
3	166	white
3	2194	white
3	1748	red
3	2388	white
3	38	white
3	496	red
3	2285	white
3	739	white
3	1482	red
3	2136	red
3	1432	white
3	1757	white
3	1257	white
3	247	red
3	1990	white
3	625	white
3	380	red
3	2266	red
3	2242	white
3	2282	red
3	1941	white
3	933	white
3	1957	red
4	1963	red
4	232	red
4	2051	white
4	1087	white
4	527	red
4	294	white
4	1731	white
4	1115	white
4	2149	white
4	344	white
4	1250	red
4	1231	white
4	339	red
4	1766	white
4	538	white
4	445	red
4	1230	white
4	1644	white
4	268	white
4	786	red
4	1532	red
4	1627	white
4	433	red
4	839	white
4	2400	white
4	2241	white
4	2220	red
4	309	white
4	2161	red
4	1529	red
4	1274	red
4	2141	white
4	2337	red
4	1551	white
4	1922	white
4	673	white
4	890	white
4	1302	white
4	2390	red
4	1093	white
4	1747	white
4	1440	white
4	286	white
4	1182	white
4	318	white
4	1719	red
4	543	red
4	1726	red
4	2	white
4	2237	white
5	1611	red
5	751	red
5	844	white
5	1002	white
5	1121	white
5	2180	white
5	2295	red
5	744	red
5	1998	red
5	775	white
5	1875	red
5	1643	white
5	523	red
5	777	white
5	274	white
5	1281	red
5	1191	white
5	1725	white
5	1504	red
5	765	white
5	525	red
5	1800	white
5	570	white
5	1206	white
5	885	white
5	2397	red
5	1052	white
5	1566	white
5	890	white
5	179	red
5	994	red
5	2096	white
5	2192	red
5	482	red
5	232	red
5	250	red
5	1523	red
5	1348	red
5	2302	red
5	1538	red
5	395	red
5	46	white
5	2341	white
5	320	white
5	2276	white
5	63	white
5	384	red
5	1519	red
5	2202	white
5	954	white
6	600	white
6	2224	red
6	1163	white
6	29	white
6	1152	white
6	1499	red
6	1031	red
6	1994	white
6	1380	white
6	100	red
6	612	red
6	1077	white
6	1459	white
6	2214	white
6	270	red
6	2084	white
6	374	white
6	2146	white
6	905	white
6	1780	red
6	913	white
6	1664	red
6	985	white
6	506	white
6	2284	red
6	1969	white
6	1275	red
6	1474	red
6	1037	red
6	232	red
6	65	white
6	2226	red
6	1421	white
6	112	white
6	78	white
6	1036	white
6	1136	white
6	1765	red
6	1347	red
6	1739	red
6	480	white
6	2366	white
6	1226	white
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6	1657	white
6	661	white
6	1023	white
6	1367	red
6	1273	white
6	217	white
7	1506	white
7	1720	white
7	1112	red
7	2291	white
7	2121	white
7	1679	white
7	1308	red
7	1985	white
7	930	red
7	2069	white
7	618	white
7	118	red
7	1978	white
7	1026	white
7	1102	red
7	1645	white
7	122	white
7	426	white
7	963	white
7	1197	white
7	74	red
7	199	white
7	2039	white
7	2018	white
7	974	white
7	1524	white
7	2344	red
7	1758	white
7	1669	white
7	1131	white
7	1733	white
7	1897	white
7	828	white
7	1343	white
7	319	white
7	807	red
7	2061	white
7	14	white
7	1024	red
7	1357	white
7	2354	white
7	129	red
7	1984	white
7	42	white
7	1861	white
7	674	red
7	1942	white
7	2178	white
7	1169	white
7	1304	red
8	1199	white
8	243	red
8	273	red
8	2092	red
8	117	white
8	766	white
8	2376	white
8	1563	white
8	1540	white
8	267	white
8	362	red
8	825	white
8	46	white
8	1609	red
8	1383	white
8	1753	red
8	51	white
8	378	white
8	1208	red
8	2196	red
8	896	red
8	2072	white
8	467	white
8	2193	red
8	1353	red
8	2205	red
8	2120	red
8	57	white
8	1916	white
8	818	red
8	312	white
8	1317	white
8	2162	red
8	1689	red
8	777	white
8	2028	white
8	658	red
8	990	white
8	67	red
8	944	white
8	1576	white
8	1645	white
8	2148	white
8	576	red
8	1250	red
8	1850	white
8	1884	red
8	920	red
8	1354	red
8	2099	white
9	2170	white
9	1107	white
9	776	white
9	2084	white
9	1503	white
9	2290	white
9	1034	red
9	1695	white
9	1933	red
9	900	white
9	2227	red
9	677	white
9	2111	white
9	1929	red
9	1031	red
9	2284	red
9	1127	white
9	2308	white
9	1609	red
9	354	red
9	189	white
9	2254	white
9	359	white
9	1725	white
9	2376	white
9	986	red
9	1308	red
9	921	red
9	1967	white
9	574	white
9	1321	white
9	2001	red
9	224	white
9	975	white
9	578	white
9	2076	red
9	2229	white
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9	1757	white
9	397	red
9	1788	red
9	244	white
9	1674	white
9	2214	white
9	1608	red
9	112	white
9	878	white
9	1928	white
9	1416	red
9	744	red
10	485	red
10	200	red
10	1556	red
10	815	red
10	1759	white
10	690	red
10	23	white
10	47	white
10	898	white
10	105	red
10	1342	white
10	1065	white
10	1772	white
10	712	white
10	2360	white
10	156	white
10	1006	white
10	1366	white
10	483	white
10	414	red
10	1355	white
10	2288	white
10	1974	red
10	209	red
10	263	white
10	1663	white
10	627	red
10	736	red
10	1668	white
10	106	white
10	1955	white
10	732	white
10	2352	white
10	305	white
10	2065	white
10	34	white
10	1519	red
10	1703	red
10	624	white
10	1089	white
10	955	red
10	293	white
10	1825	white
10	401	white
10	1719	red
10	184	white
10	189	white
10	1440	white
10	589	red
10	592	white
11	783	white
11	751	red
11	161	white
11	1538	red
11	222	red
11	971	red
11	418	red
11	1800	white
11	1433	white
11	1275	red
11	503	white
11	768	white
11	1543	white
11	1205	red
11	413	red
11	110	white
11	2043	red
11	561	white
11	2034	white
11	763	white
11	726	red
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11	1787	white
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11	296	white
11	2132	white
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11	1241	white
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11	2110	white
11	2349	red
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11	248	white
11	821	red
11	777	white
11	2372	white
11	1330	white
11	654	white
11	11	white
11	526	white
11	1983	white
11	166	white
11	612	red
11	1455	white
11	761	red
11	596	red
11	464	white
12	1709	red
12	232	red
12	1301	white
12	609	white
12	195	white
12	1304	red
12	804	white
12	2262	red
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12	1110	white
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12	1523	red
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12	1851	red
12	1006	white
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12	2036	white
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12	1130	white
12	1656	white
12	306	red
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12	2067	red
13	600	white
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13	1981	red
13	821	red
13	1898	white
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13	1873	red
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14	632	red
14	205	white
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14	512	white
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Observe that while the first 50 rows of replicate are equal to 1, the next 50 rows of replicate are equal to 2. This is telling us that the first 50 rows correspond to the first sample of 50 balls while the next 50 correspond to the second sample of 50 balls. This pattern continues for all reps = 30 replicates and thus virtual_samples has \(30 \times 50 = 1500\) rows.

virtual_prop_red <- virtual_samples %>% 
  group_by(replicate) %>% 
  summarize(red = sum(color == "red")) %>% 
  mutate(prop_red = red / 50)

virtual_prop_red

# A tibble: 30 x 3
   replicate   red prop_red
       <int> <int>    <dbl>
 1         1    20     0.4 
 2         2    21     0.42
 3         3    23     0.46
 4         4    18     0.36
 5         5    24     0.48
 6         6    19     0.38
 7         7    12     0.24
 8         8    23     0.46
 9         9    19     0.38
10        10    16     0.32
# … with 20 more rows

#kable(virtual_prop_red) # To see all 30 samples

Let’s visualize the distribution of these 33 proportions red based on 33 virtual samples using a histogram with binwidth = 0.05

ggplot(virtual_prop_red, aes(x = prop_red)) +
  geom_histogram(binwidth = 0.05, boundary = 0.4, color = "white", fill = "steelblue") +
  labs(x = "Proportion of 50 balls that were red", 
       title = "Distribution of 30 proportions red") 

Observe that occasionally we obtained proportions red that are less than ____, while on the other hand we occasionally we obtained proportions that are greater than ____. However, the most frequently occurring proportions red out of 50 balls were between ____ % and ____ % (for ___ out 30 samples). Why do we have these differences in proportions red? Because of ___________________.

Exercise 1.1 Redo the above activity with 1000 repeated samples and state your conclusions.

1.2.1 Using different shovels

If your goal was still to estimate the proportion of the bowl’s balls that were red, which shovel would you choose? Why? Let’s try to answer these questions.

# Segment 1: sample size = 25 ------------------------------
# 1.a) Virtually use shovel 1000 times
virtual_samples_25 <- bowl %>% 
  rep_sample_n(size = 25, reps = 1000)

# 1.b) Compute resulting 1000 replicates of proportion red
virtual_prop_red_25 <- virtual_samples_25 %>% 
  group_by(replicate) %>% 
  summarize(red = sum(color == "red")) %>% 
  mutate(prop_red = red / 25)

# 1.c) Plot distribution via a histogram
p1 <- ggplot(virtual_prop_red_25, aes(x = prop_red)) +
  geom_histogram(binwidth = 0.05, boundary = 0.4, color = "white") +
  labs(x = "Pro of 25 balls that were red", title = "25") 

# Segment 2: sample size = 50 ------------------------------
# 2.a) Virtually use shovel 1000 times
virtual_samples_50 <- bowl %>% 
  rep_sample_n(size = 50, reps = 1000)

# 2.b) Compute resulting 1000 replicates of proportion red
virtual_prop_red_50 <- virtual_samples_50 %>% 
  group_by(replicate) %>% 
  summarize(red = sum(color == "red")) %>% 
  mutate(prop_red = red / 50)

# 2.c) Plot distribution via a histogram
p2 <- ggplot(virtual_prop_red_50, aes(x = prop_red)) +
  geom_histogram(binwidth = 0.05, boundary = 0.4, color = "white") +
  labs(x = "Pro of 50 balls that were red", title = "50")  

# Segment 3: sample size = 100 ------------------------------
# 3.a) Virtually using shovel with 100 slots 1000 times
virtual_samples_100 <- bowl %>% 
  rep_sample_n(size = 100, reps = 1000)

# 3.b) Compute resulting 1000 replicates of proportion red
virtual_prop_red_100 <- virtual_samples_100 %>% 
  group_by(replicate) %>% 
  summarize(red = sum(color == "red")) %>% 
  mutate(prop_red = red / 100)

# 3.c) Plot distribution via a histogram
p3 <- ggplot(virtual_prop_red_100, aes(x = prop_red)) +
  geom_histogram(binwidth = 0.05, boundary = 0.4, color = "white") +
  labs(x = "Pro of 100 balls that were red", title = "100") 


plot_grid(p1, p2, p3, nrow = 1)

Observe that as the sample size increases, the ______ of the 1000 replicates of the proportion red decreases. In other words, as the sample size increases, there are less differences due to sampling variation and the distribution centers more tightly around the same value. Eyeballing the above Figure, things appear to center tightly around roughly ____%.

# n = 25
virtual_prop_red_25 %>% 
  summarize(sd = sd(prop_red))

# A tibble: 1 x 1
      sd
   <dbl>
1 0.0957

# n = 50
virtual_prop_red_50 %>% 
  summarize(sd = sd(prop_red))

# A tibble: 1 x 1
      sd
   <dbl>
1 0.0702

# n = 100
virtual_prop_red_100 %>% 
  summarize(sd = sd(prop_red))

# A tibble: 1 x 1
      sd
   <dbl>
1 0.0486

Number of slots in shovel	Standard deviation of proportions red
25	0.0978
50	0.0669
100	0.0471

As the sample size increases our numerical measure of spread decreases; there is less variation in our proportions red. In other words, as the sample size increases, our guesses at the true proportion of the bowl’s balls that are red get more consistent and precise.

1.3 What did we learn?

This was our first attempt at understanding two key concepts relating to sampling for estimation:

1. The effect of sampling variation on our estimates.
2. The effect of sample size on sampling variation.

Let’s now introduce some terminology and notation as well as statistical definitions related to sampling.

1.4 Terminology & notation

1. (Study) Population: A (study) population is a collection of individuals or observations about which we are interested. We mathematically denote the population’s size using upper case N. In our simulations the (study) population was the collection of N = 2400 identically sized red and white balls contained in the bowl.

2. Population parameter: A population parameter is a numerical summary quantity about the population that is unknown, but you wish you knew. For example, when this quantity is a mean, the population parameter of interest is the population mean which is mathematically denoted with the Greek letter \(\mu\) (pronounced “mu”). In our simulations however since we were interested in the proportion of the bowl’s balls that were red, the population parameter is the population proportion which is mathematically denoted with the letter \(p\).

3. Census: An exhaustive enumeration or counting of all \(N\) individuals or observations in the population in order to compute the population parameter’s value exactly. In our simulations, this would correspond to manually going over all \(N = 2400\) balls in the bowl and counting the number that are red and computing the population proportion \(p\) of the balls that are red exactly. When the number \(N\) of individuals or observations in our population is large, as was the case with our bowl, a census can be very expensive in terms of time, energy, and money.

4. Sampling: Sampling is the act of collecting a sample from the population when we don’t have the means to perform a census. We mathematically denote the sample’s size using lower case \(n\), as opposed to upper case \(N\) which denotes the population’s size. Typically the sample size \(n\) is much smaller than the population size \(N\), thereby making sampling a much cheaper procedure than a census. In our simulations, we used shovels with 25, 50, and 100 slots to extract a sample of size \(n = 25\), \(n = 50\), and \(n = 100\) balls.

5. Point estimate (AKA sample statistic): A summary statistic computed from the sample that estimates the unknown population parameter. In our simulations, recall that the unknown population parameter was the population proportion and that this is mathematically denoted with p. Our point estimate is the sample proportion: the proportion of the shovel’s balls that are red. In other words, it is our guess of the proportion of the bowl’s balls balls that are red. We mathematically denote the sample proportion using \(\hat{p}\); the “hat” on top of the p indicates that it is an estimate of the unknown population proportion \(p\).

6. Representative sampling: A sample is said be a representative sample if it is representative of the population. In other words, are the sample’s characteristics a good representation of the population’s characteristics? In our simulations, are the samples of \(n\) balls extracted using our shovels representative of the bowl’s \(N = 2400\) balls?

7. Generalizability: We say a sample is generalizable if any results based on the sample can generalize to the population. In other words, can the value of the point estimate be generalized to estimate the value of the population parameter well? In our simulations, can we generalize the values of the sample proportions red of our shovels to the population proportion red of the bowl? Using mathematical notation, is \(\hat{p}\) a “good guess” of \(p\)?

8. Bias: In a statistical sense, we say bias occurs if certain individuals or observations in a population have a higher chance of being included in a sample than others. We say a sampling procedure is unbiased if every observation in a population had an equal chance of being sampled. In our simulations, since each ball had the same size and hence an equal chance of being sample in our shovels, our samples were unbiased.

9. Random sampling: We say a sampling procedure is random if we sample randomly from the population in an unbiased fashion. In our simulations, this would correspond to sufficiently mixing the bowl before each use of the shovel.

Let’s put them all together:

• If we extract a sample of \(n=50\) balls at random, in other words we mix the equally-sized balls before using the shovel, then

• the contents of the shovel are an unbiased representation of the contents of the bowl’s 2400 balls, thus

• any result based on the sample of balls can generalize to the bowl, thus

• the sample proportion \(\hat{p}\) of the \(n=50\) balls in the shovel that are red is a “good guess” of the population proportion \(p\) of the \(N =2400\) balls that are red, thus

• instead of manually going over all the balls in the bowl, we can infer about the bowl using the shovel.

Definition 1.1 The sampling distribution of a Statistic (e.g. Mean, Median, Proportion, etc) is its probability distribution.

Definition 1.2 The standard deviation of a sampling distribution is called the standard error.

Example: This is the same table as above, but notice the 2nd column name.

Number of slots in shovel	Standard Error of proportions red
25	0.0978
50	0.0669
100	0.0471

Exercise 1.2 Find and plot the sampling distribution of the proportion (\(\hat{p}\)) of heads when you flip a fair coin. (Use 5000 sets of 10 tosses)

1. What is the sample size?

2. How many experiments?

3. Find and plot the sampling distribution of \(\hat{p}\).

4. Find the standard error of the sampling distribution of \(\hat{p}\).

5. What happens to the standard error, when you increase the sample size?

---

1.5 DETOUR — Some brand name distributions

1.5.1 Normal Distribution

The normal distribution is defined by the following probability density function, where \(\mu\) is the population mean and \(\sigma\) is the standard deviation.

\[f(x) = \dfrac{1}{\sigma \sqrt{2 \pi}}e^{-(x-\mu)^2/{2\sigma^2}}\]

If a random variable \(X\) follows the normal distribution, then we write: \(X \sim N(\mu, \sigma^2)\)

Here is how the normal density looks like: Ex: here \(X \sim N(0, 1)\)

# Ignore this code

p1 <- ggplot(data = data.frame(x = c(-3, 3)), aes(x)) +
  stat_function(fun = dnorm, n = 101, args = list(mean = 0, sd = 1)) + ylab("") +
  scale_y_continuous(breaks = NULL)
p1

R functions: (Package: stats)

Examples:

1. Generate a random sample of 100 from \(N(15, 9)\) and create a histrogram.

x <- rnorm(n = 100, mean = 15, sd = 3)

ggplot(data.frame(x), aes(x = x)) + geom_histogram(binwidth = 1.5) 

2. If \(X \sim N(15, 9)\) find the probability that X being greater than 21: \(P(X > 21)\)

pnorm(21, mean = 15, sd = 3) # pnorm gives us the left tail area to a given number, 21 in this case

[1] 0.9772499

3. If \(X \sim N(15, 9)\) find the 25th quantile.

qnorm(.25, mean = 15, sd = 3)

[1] 12.97653

#pnorm(12.97653, mean = 15, sd = 3)

Question:

1. A radar unit is used to measure speeds of cars on a motorway. The speeds are N(90 km/hr, 10 km/hr). What is the probability that a car picked at random is travelling at more than 100 km/hr?

2. GMAT are roughly normally distributed with a mean of 527 and a standard deviation of 112. How high must an individual score on the GMAT in order to score in the highest 5%?

1.5.2 Exponential Distribution

The Exponential Distribution is defined by the following probability density function, where \(\dfrac{1}{\lambda}\) is the population mean and standard deviation.

\[f(x) = \lambda e^{-\lambda x}\]

If a random variable \(X\) follows the Exponential Distribution, then we write: \(X \sim Exp(\lambda)\)

Here is how the Exponential density looks like: Ex: here \(X \sim Exp(1/15)\)

# Ignore this code
x <- seq(0, 100, length.out=1000)
dat <- data.frame(x=x, px=dexp(x, rate=1/15))

ggplot(dat, aes(x=x, y=px)) + geom_line()

R functions: (Package: stats)

Examples:

1. Generate a random sample of 100 from \(Exp(1/15)\) and create a histrogram.

x <- rexp(n = 100, rate = 1/15)

ggplot(data.frame(x), aes(x = x)) + geom_histogram(binwidth = 5) 

2. If \(X \sim Exp(1/15)\) find the probability that X being less than 21: \(P(X > 21)\)

pexp(21, rate = 1/15)

[1] 0.753403

3. If \(X \sim Exp(1/15)\) find the 75th quantile.

qexp(.75, rate = 1/15)

[1] 20.79442

#pexp(20.79442, rate = 1/15)

Question:

The number of days ahead travelers purchase their airline tickets can be modeled by an exponential distribution with the average amount of time equal to 15 days.

1. Find the probability that a traveler will purchase a ticket fewer than ten days in advance.

2. How many days do 80% of all travelers wait?

1.5.3 Binomial Distribution

The binomial distribution is a discrete probability distribution. It describes the outcome of n independent trials in an experiment. Each trial is assumed to have only two outcomes, either success or failure. If the probability of a successful trial is p, then the probability of having x successful outcomes in an experiment of n independent trials is as follows.

\[f(x) = {n \choose x} p^x (1-p)^{(n-x)} \quad \text{where x = 0, 1, 2,...,n}\]

Example: Suppose there are twelve multiple choice questions in an English class quiz. Each question has five possible answers, and only one of them is correct.

1. Find the probability of having exactly four correct answers if a student attempts to answer every question at random.

2. Find the probability of having four or less correct answers if a student attempts to answer every question at random.

Solution:

Since only one out of five possible answers is correct, the probability of answering a question correctly by random is 1/5=0.2.

1. By hand:

\({12 \choose 4} 0.2^4 (1-0.2)^{(12-4)} = 0.1329\)

In R:

dbinom(4, size=12, prob=0.2) 

[1] 0.1328756

2. By hand:

\({12 \choose 4} 0.2^4 (1-0.2)^{(12-4)} + {12 \choose 3} 0.2^3 (1-0.2)^{(12-3)} + {12 \choose 2} 0.2^2 (1-0.2)^{(12-2)} + {12 \choose 1} 0.2^1 (1-0.2)^{(12-1)} + {12 \choose 0} 0.2^0 (1-0.2)^{(12-0)} = 0.9274\)

In R:

OR Alternatively,

END of the DETOUR!

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1.6 Theoritical Sampling Distribution of the Proportion

Sampling distribution of \(\hat{p}\) is normal with mean \(p\)(Actual population proportion) and standard deviation of \(\sqrt{p(1-p)/n}\)

\[\hat{p} \sim N( p, \sqrt{p(1-p)/n} )\]

Let’s revisit the coin flip example: What we did is an simulation. Now we can check how close our simulation results are to the theoritical results.

Now let’s find the theoritical mean and the standard deviation the sampling distribution of the proportion:

Here \(p = 0.5\), \(n = 10\)

Therefore \(\hat{p} \sim N( p, \sqrt{p(1-p)/n} ) = N( 0.5, \sqrt{0.5(1-0.5)/10} ) = N( 0.5, 0.1581139)\)

1.7 Sampling Distribution of the Mean